ABERRATION IN A DISPERSIVE MEDIUM 43
when Ğ;=(), or a multiple of 2?r ; but when ie is an odd multiple of TT, there is a reversal of sign, equivalent to a change of phase of half a period. And the places where these particular effects occur travel along the screen with a velocity a which is supposed to be small relatively to that of light. In the absence of the screen the luminous vibration is represented by
(j> = cos (nt kz\ .............................. (1)
or at. the place of (lie screen, where z = 0, by
<jb = cos nt simply.
In accordance with the suppositions already made, the vibration just behind the screen will be
<j) = cos m (vt ,'/?) . cos nt | cos {(n + mv) t inn;} 4- $ cos {(n ~ 'inv) t + mos] ; ...... (2)
and the question is to find what form <j> will take at a finite distance z behind the, screen.
It 5s not difficult to see that for this purpose we have only to introduce terms proportional to z into the arguments of the cosines. Thus, if we write
<jb =s ^ C.OH {(//. + mv) t in..r. ~ /^ z\ + ^ cos |(?t niv) t + moo fj,zz}, . . .(8)
\vc may determine filt ^ HO as to satisfy in each case the general differential equation of propagation, vlx.
In (4) V IM constant when the medium in noti-diapersivc ; but in the contmry MW V must be givun different valueH, Bay Vl and F"a, when the coeffici(!ut of t IH it+mv or umv. Thus
(n 4- //in)1 * P? (-//iğ + wta), (n - mv)ğ Ğ Fg* (ma 4- 7n9a) ....... (5)
The coefficients /tj, y^ being determined in accordance with (5), the value of $ in (8) natinfioB all the requirements of the problem. It may also be written
(f> -ğ COM [mvt mte - |- (^ /*a) *} GOB [ni ^ (^ + /jt,2)z}, ...... (6)
of which the first factor, varying slowly with t, may be regarded as the amplitude of the luminous vibration.
The condition of constant amplitude at a given time is that mso+^^i ^ z Khali remain unchanged. Thus the amplitude which is to be found at #=0 on the screen prevails also behind the screen along the line
so that (7) may be regarded as the angle of aberration due to v. It remains to express this angle by means of (5) in terms of the fundamental data.approximately given wave-length.